Asymptotics of Higgs bundles in the Hitchin component

Collier, Brian; Li, Shuyu

doi:10.1016/j.aim.2016.11.031

Cited by 21 publications

(39 citation statements)

References 37 publications

(51 reference statements)

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“…Some cases were verified in [3]. (See [3,7] for the precise statements.) We emphasize that we can obtain this kind of estimate for more general families of harmonic bundles given on complex curves which are not necessarily compact.…”

Section: Hitchin Wkb-problemmentioning

confidence: 99%

“…This study has grown out of my effort to understand (part of) the intriguing works [3], [5], [7] and [11,12]. I thank Carlos Simpson for inspiring discussions on many occasions.…”

Section: Acknowledgementmentioning

confidence: 99%

“…Under some assumptions on the spectral curves, it looks possible to generalize the method in [11,12] and our method in this paper. See also [3] for some Toda-like cases.…”

Section: Limiting Configurationmentioning

confidence: 99%

“…We explain a singular perturbation theory which is available in our situation, and which seems slightly different from those in [3] and [7]. Note that we applied Corollary 2.19 in the proof of Theorem 2.17 to find a family of small gauge transforms with which the family of connections γ * D on [0, 1] are transformed to the connections whose connection matrices are diagonal.…”

Section: Appendix: a Singular Perturbation Theorymentioning

confidence: 99%

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Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces

Mochizuki¹

2016

J Topology

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Let (E, ∂E, θ) be a stable Higgs bundle of degree 0 on a compact connected Riemann surface. Once we fix a flat metric h det(E) on the determinant of E, we have the harmonic metrics ht (t > 0) for the stable Higgs bundles (E, ∂E, tθ) such that det(ht) = h det(E) . We study the behaviour of ht when t goes to ∞. First, we show that the Hitchin equation is asymptotically decoupled under the assumption that the Higgs field is generically regular semisimple. We apply it to the study of the so called Hitchin WKB-problem. Second, we study the convergence of the sequence (E, ∂E, θ, ht) in the case rank E = 2. We introduce a rule to determine the parabolic weights of a "limiting configuration", and we show the convergence of the sequence to the limiting configuration in an appropriate sense. The results can be appropriately generalized in the context of Higgs bundles with a Hermitian-Einstein metric on curves.

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Section: Hitchin Wkb-problemmentioning

confidence: 99%

“…This study has grown out of my effort to understand (part of) the intriguing works [3], [5], [7] and [11,12]. I thank Carlos Simpson for inspiring discussions on many occasions.…”

Section: Acknowledgementmentioning

confidence: 99%

“…Under some assumptions on the spectral curves, it looks possible to generalize the method in [11,12] and our method in this paper. See also [3] for some Toda-like cases.…”

Section: Limiting Configurationmentioning

confidence: 99%

Section: Appendix: a Singular Perturbation Theorymentioning

confidence: 99%

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Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces

Mochizuki¹

2016

J Topology

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“…For instance compactifications of spaces of representations of Γ have been introduced and studied in [Par12,Ale08,Le12]. In the context of Hitchin representations, asymptotic properties of diverging sequences are studied in [Zha13,Zha14,CL14,Lof15,Par15,KNPS15,MSWW14].…”

Section: Introductionmentioning

confidence: 99%

Maximal representations, non-Archimedean Siegel spaces, and buildings

Burger

Pozzetti

2017

Geom. Topol.

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Abstract. Let F be a real closed field. We define the notion of a maximal framing for a representation of the fundamental group of a surface with values in Sp(2n, F). We show that ultralimits of maximal representations in Sp(2n, R) admit such a framing, and that all maximal framed representations satisfy a suitable generalization of the classical Collar Lemma. In particular this establishes a Collar Lemma for all maximal representations into Sp(2n, R). We then describe a procedure to get from representations in Sp(2n, F) interesting actions on affine buildings, and, in the case of representations admitting a maximal framing, we describe the structure of the elements of the group acting with zero translation length.

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Constructing Buildings and Harmonic Maps

Katzarkov

Noll²,

Pandit

et al. 2017

Algebra, Geometry, and Physics in the 21st Century

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In a continuation of our previous work [17], we outline a theory which should lead to the construction of a universal pre-building and versal building with a φ-harmonic map from a Riemann surface, in the case of two-dimensional buildings for the group SL 3 . This will provide a generalization of the space of leaves of the foliation defined by a quadratic differential in the classical theory for SL 2 . Our conjectural construction would determine the exponents for SL 3 WKB problems, and it can be put into practice on examples. P B .

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Asymptotics of Higgs bundles in the Hitchin component

Cited by 21 publications

References 37 publications

Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces

Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces

Maximal representations, non-Archimedean Siegel spaces, and buildings

Constructing Buildings and Harmonic Maps

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